Math / Geometry

QuestionWhat is the area of this composite figore?

Studdy Solution

STEP 1

1. The semicircle has a diameter of $6$ meters.
2. The rectangle above the semicircle is $6$ meters by $6$ meters.
3. The larger rectangle on top is $12$ meters by $8$ meters.
4. The composite figure is the sum of the areas of these three parts.

STEP 2

1. Calculate the area of the semicircle.
2. Calculate the area of the smaller rectangle.
3. Calculate the area of the larger rectangle.
4. Sum the areas to find the total area of the composite figure.

STEP 3

Calculate the area of the semicircle.
The formula for the area of a circle is:
$A = \pi r^2$
Since the semicircle has a diameter of $6$ meters, the radius $r$ is:
$r = \frac{6}{2} = 3 \text{ meters}$
The area of the semicircle is half the area of a full circle:
$A_{\text{semicircle}} = \frac{1}{2} \pi (3)^2$ $= \frac{1}{2} \pi \times 9$ $= \frac{9}{2} \pi$

STEP 4

Calculate the area of the smaller rectangle.
The formula for the area of a rectangle is:
$A = \text{length} \times \text{width}$
For the smaller rectangle:
$A_{\text{small rectangle}} = 6 \times 6$ $= 36 \text{ square meters}$

STEP 5

Calculate the area of the larger rectangle.
For the larger rectangle:
$A_{\text{large rectangle}} = 12 \times 8$ $= 96 \text{ square meters}$

STEP 6

Sum the areas to find the total area of the composite figure.
$A_{\text{total}} = A_{\text{semicircle}} + A_{\text{small rectangle}} + A_{\text{large rectangle}}$ $= \frac{9}{2} \pi + 36 + 96$
Assuming $\pi \approx 3.14159$ :
$A_{\text{total}} \approx \frac{9}{2} \times 3.14159 + 36 + 96$ $\approx 14.1372 + 36 + 96$ $\approx 146.1372 \text{ square meters}$
The approximate total area of the composite figure is:
$\boxed{146.14 \text{ square meters}}$

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